ISTANBUL TECHNICAL UNIVERSITY F INSTITUTE OF SCIENCE AND TECHNOLOGY

A SEMICLASSICAL KINETICTHEORY OF THE DIRAC PARTICLES

M.Sc. THESIS

Eda KILINÇARSLAN

Department of Physics Engineering

Physics Engineering Programme

MAY 2015

ISTANBUL TECHNICAL UNIVERSITY F INSTITUTE OF SCIENCE AND TECHNOLOGY

A SEMICLASSICAL KINETICTHEORY OF THE DIRAC PARTICLES

M.Sc. THESIS

Eda KILINÇARSLAN(509131102)

Department of Physics Engineering

Physics Engineering Programme

Thesis Advisor: Prof. Dr. Ömer F. DAYI

MAY 2015

ISTANBUL TEKNIK ÜNIVERSITESI F FEN BILIMLERI ENSTITÜSÜ

DIRAC PARÇACIKLARININYARI KLASIK KINETIK KURAMI

YÜKSEK LISANS TEZI

Eda KILINÇARSLAN(509131102)

Fizik Mühendisligi Bölümü

Fizik Mühendisligi Programı

Tez Danısmanı: Prof. Dr. Ömer F. DAYI

MAYIS 2015

Eda KILINÇARSLAN, a M.Sc. student of ITU Institute of Science and Technology509131102 successfully defended the thesis entitled “A SEMICLASSICAL KINETICTHEORY OF THE DIRAC PARTICLES”, which he/she prepared after fulfilling therequirements specified in the associated legislations, before the jury whose signatures arebelow.

Thesis Advisor : Prof. Dr. Ömer F. DAYI ..............................Istanbul Technical University

Jury Members : Prof. Dr. Teoman TURGUT ..............................Bogazici University

Doç. Dr. A. Levent SUBASI ..............................Istanbul Technical University

Date of Submission : 30 April 2015Date of Defense : 26 May 2015

v

to Berkin Elvan,

vii

FOREWORD

I am sincerely indebted to my supervisior Ömer F. Dayı for sharing precious ideas withme and for help whenever I need to complete this thesis.

Special thanks to Fatih Usta for his moral support in my hard times during this work.

May 2015 Eda KILINÇARSLAN(Physics Engineer)

ix

x

TABLE OF CONTENTS

Page

FOREWORD................................................................................................................ ixTABLE OF CONTENTS............................................................................................. xiABBREVIATIONS ...................................................................................................... xiiiSUMMARY .................................................................................................................. xvÖZET ............................................................................................................................xvii1. Introduction.............................................................................................................. 12. Semiclassical Diagonalization of The Dirac Hamiltonian.................................... 53. Wave Packet Formalism of the First Order Lagrangian ..................................... 74. Semiclassical Dynamics of the Dirac Particles...................................................... 95. Distribution Function and Continuity Equation ................................................. 156. Massless Fermions ................................................................................................... 197. Thomas Precession................................................................................................... 218. Time Evolution of Spin............................................................................................ 259. Results and Discussion ............................................................................................ 27REFERENCES............................................................................................................. 29APPENDICES.............................................................................................................. 31

Berry Gauge Fields.................................................................................................... 33CURRICULUM VITAE.............................................................................................. 35

xi

xii

ABBREVIATIONS

GBM : Gosselin-Berard-MohrbachBMT : Bargmann-Michel-Teledgi

xiii

xiv

A SEMICLASSICAL KINETICTHEORY OF THE DIRAC PARTICLES

SUMMARY

The semiclassical kinetic theory of massive spin-12 particles interacting with the external

electromagnetic fields is formulated in terms of differential forms which are matrix valuedin spin space. Semiclassical approximation is performed by employing the wave packetconstructed as superposition of positive energy plane wave solutions of the free Diracequation. A symplectic two-form is derived using the wave packet. It is a matrixin “spin indices” and possesses a term related to the Berry curvature obtained from anon-Abelian Berry gauge field. Time evolution of phase space variables in terms ofphase space themselves are attained by making use of the volume form which is alsoa matrix. Continuity equation for particle number density and the particle current densityare obtained by introducing a change of basis in order to define distribution functionsin the helicity basis. The massless limit is derived by constructing the helicity statesexplicitly.

When one deals with a non-relativistic formulation of massive particles the equationsof motion should be corrected with a relativistic kinematic factor known as Thomasprecession. Its origin lies in the fact that when one would like to write two successiveLorentz boost as one Lorentz boost it should be accompanied with a rotation whose angledepends on the related velocities. It is shown that Thomas precession can be includedstraightforwardly into the semiclassical formulation adopted in the thesis. It alters theequations of motion and cancels the anomalous velocity terms appearing due to the Berrycurvature.

Initially, I will derive the semiclassical block diagonal Hamiltonian for a Dirac particlein the electromagnetic field including all terms at the first order in Planck constant usingthe Gosselin-Berard-Mohrbach method. In this method the unitary transformation whichblock diagonalizes the Hamiltonian possesses terms related to the Berry gauge fields.In general curvature of the Berry gauge fields appear as the phase factor of a quantumstate transported adiabatically. When the block diagonalization is carried out by unitarytransformation, the dynamical operators should also be transformed and they becomenon-commutative. I will use these non-commutative phase space operators to derive thetime evolution of spin matrices which will be introduced in the course of finding thesemiclassical formulation.

The one-form corresponding to first order Lagrangian is defined by making use of thewave packet built with the positive energy solutions of the Dirac equation. This one-formcan be written as a matrix whose indices correspond to the positive energy solutions whichare called spin indices. It has a term depending on the non-Abelian Berry gauge fieldsgiven by the degenerate positive energy solutions. Then the symplectic two-form derivedfrom this one-form includes a term which depends on the Berry curvature. I use thedifferential form formalism to obtain the equations of motion of phase space variables.A straightforward method is applied to find solutions of the equations of motion for the

xv

phase space velocities in terms of the phase space variables employing Liouville equationand the differential form formalism.

To get the kinetic theory of Dirac particles I need the distribution function which canbe used to define the particle number density. However, it is a matrix whose elementsshould be interpreted appropriately. The mostly adopted procedure is to choose a specificconfiguration where the third component of spin is a conserved quantity. Then onecan set the off-diagonal terms to zero. In general spin is not a conserved quantitybut helicity operator gives a vanishing commutator with the free Dirac Hamiltonian.Moreover when I discuss the massless limit it would be essential to split the right-handedand the left-handed contributions. Therefore, the appropriate basis is the one where thehelicity operator is diagonal. I define this new basis and obtain the continuity equationfor the Dirac particle using the distribution function which is diagonal. Then I derivethe continuity equation for the particle number density and the particle number currentdensity. Obviously, because of possessing the solutions of the equations of motion for thevelocities in terms of phase space variables one can directly obtain the particle current.

Obtaining the massless limit in the helicity basis is straightforward. It yields the continuityequation which has an anomaly term. The particle current possesses an anomalousvelocity term and a term leading to the chiral magnetic effect.

Thomas precession which shows up as the relativistic correction in the equations ofmotion are obtained. I briefly discuss what is the source of the Thomas precession. Then Ipresent how one should introduce this correction into the wave packet formalism. It givesa contribution to the initial one-form on the same footing with the Berry gauge field. Infact up to higher order terms in momentum it gives the opposite contribution of the Berrygauge field and cancels the anomalous velocity terms given by the Berry curvature. Thisresult coincides with the ones obtained within the relativistic formulations of the Diracparticles.

Originally the Thomas precession is used to obtain the corrections to the non-relativisticformulation of the time evolution of spin matrices. However, the formalism which Iadopted is not aware of the time evolution of spin. For completeness I show that it canbe integrated into the formalism by making use of the non-commutative charter of thedynamical variables obtained in the Gosselin-Berard-Mohrbach method. Time evolutionof spin matrices are shown to be the same with the Bargmann-Michel-Telegdi equation.

Lastly, the results obtained in the thesis and the possible extensions are discussed.

xvi

DIRAC PARÇACIKLARININYARI KLASIK KINETIK KURAMI

ÖZET

Kütleli spin-1/2 parçacıkların hareketini ifade etmek için Dirac denklemi kullanılır. Ikisipozitif enerji, ikisi de negatif enerji çözümlerine ait olmak üzere Dirac denkleminin dörttane çözümü vardır. Kuantum mekaniksel olarak parçacıkların hareketi dalga paketikurularak ifade edilmektedir. Fakat negatif enerji çözümlerinin de olması parçacıkyorumunu zorlastırır. Bu nedenle relativistik olmayan yarı klasik dinamigi, Diracdenkleminin pozitif enerji çözümlerini içeren dalga paketi olusturarak elde edecegim.

Yarı klasik limit bazı kuantum mekaniksel etkileri daha iyi anlamak için yararlıolabilecek bir yöntemdir. Dirac denkleminde kütlesiz limite gidildiginde kiral yada Weyl parçacıgı adı verilen parçacıkların hareketini ifade eden denklem eldeedilmis olur. Son zamanlardaki çalısmalarda, 3 + 1 boyutta, dıs elektromanyetikalan nedeniyle olusan anomali terimlerinin kiral parçacıklarının yarı klasik kinetikkuramında nasıl yerlestirilebilecegi gösterilmistir. Yüksek boyutta genellestirilmishareket denklemlerinin, faz uzayı degiskenlerine baglı çözümlerini de tam olarak verenyöntem matris degerlidir. Bu yöntemde digerlerinden farklı olarak faz uzayı degiskenlerikonum ve momentumdur, spine karsı gelen klasik bir nicelik yoktur. Klasik faz uzayıdegiskenleri matris olmadıkları halde hareket denklemleri ile bulunan hız degiskenlerimatris degerlidir. Matrislerin “spin indisleri” farklı pozitif çözümlere karsılık gelir.

Kiral kinetik kuramının yarı klasik formalizminin en önemli bilesenlerinden biri kuantummekaniksel bir faz faktörü olan Berry fazını veren Berry ayar alanlarıdır. Kuantummekaniksel bir sistemde, Hamilton yogunlugunun baglı oldugu dıs parametreler, bunlarelektrik ve manyetik alan olarak düsünülebilir, çok yavas degistirildiginde adiyabatikkurama göre, sistemin kuantum durumu degismez. Burada bahsedilen yavas degisim,parametrelerin çevrimsel bir egri üzerinde hareket etmesi olarak ifade edilebilir. Bukapalı egri tamamlandıgında sistemdeki durum vektörü bir faz kazanır. Bu faz dinamik vegeometrik iki kısımdan olusmaktadır. Adiyabatik degisim altındayken, durum vektörününkazandıgı geometrik faz çarpanına Berry fazı denir. Berry ayar alanlarının egriligikuantum mekaniksel bir faz çarpanı olan Berry fazını verir.

Bu yöntemle birinci derece Lagrange yogunlugu ile yarıklasik Hamilton dinamigini eldeetmek için ilk olarak yarı klasik blok kösegen Hamilton yogunlugu verilmelidir. Yarıklasik Hamilton yogunlugunu kösegenlestirecek uniter matris dönüsümünün Planck sabitibölü 2π , h, ye göre birinci mertebe tüm terimleri verecek sekilde yazılması gerekir.

Dalga paketi ve diferansiyel formlar kullanılarak Dirac parçacıklarının yarı klasikkinetik kuramı elde edilebilir. Bu yöntemin bazı avantajları bulunmaktadır. Öncelikle,formalizmde spin özgürlük derecesine karsılık gelen klasik bir nicelik bulunmadıgı içinhareket denklemlerinin faz uzayı hızlarının faz degiskenleri cinsinden veren çözümleriaçıkça bulunabilir. Böylece parçacık akısı rahatlıkla yazılabilir.

Tez kapsamında, öncelikle elektromanyetik alan içindeki kütleli spin-1/2 parçacıklariçin Dirac Hamilton yogunlugunu blok kösegen hale getirecegim. Dalga paketi ve

xvii

diferansiyel form yöntemiyle hareket denklemlerini ve çözümlerini bulacagım. Buradafaz uzayı degiskenlerinin hız denklemi çözümleriyle hesaplanan, hız denklemlerininfaz uzayı degiskenlerine baglı çözümleridir. Bu denklemlerin elde edilisi sırasındadenklemlerimizin içine Berry ayar alanlarının girdigini görecegiz. Daha sonra yarı klasikDirac parçacıkları için dagılım fonksiyonunu uygun bazda yazıp süreklilik denkleminibulacagım. Kütleli fermiyonlar için elde ettigim hareket denklemlerinin çözümlerindekütlesiz limite giderek ve baz degistirerek elektromanyetik alandaki Weyl parçacıklarınınhareket denklemlerinin çözümlerine ulasacagım. Thomas presesyonun ne oldugunuanlatıp inceledigim yarı klasik formalizmde Thomas presesyonunun elde ettigim hızdenklemlerine nasıl bir katkı yapacagını inceleyecegim. Son olarak ise elektromanyetikalan altındaki yarı klasik Dirac parçacıgının spinin zaman içindeki degisimini verendenklemi bulacagım.

Ilk kısımda, Gosselin-Berard-Mohrbach yöntemini kullanarak elektromanyetik alandahareket eden kütleli parçacıklar için yarı klasik blok kösegen Hamilton yogunlugunuhesaplayacagım. Bu yöntem birkaç adımdan olusmaktadır. Hesaplanacak olan Hamiltonyogunlugunun h ye göre birinci dereceden olan tüm terimleri içermesini istiyorum.Elektromanyetik alan altındaki Hamilton yogunlugunun baglı oldugu degiskenleri, xxx veppp yi, birbirleriyle komute edecek sekilde aldıgınızda Hamilton yogunlugu klasik birbüyüklüktür olur. Klasik Hamiltonyeni kösegenlestirmek için uniter Foldy-Wouthuysendönüsümleri, UFW , kullanılır. Yarı klasik blok kösegen Hamilton yogunlugunuhesaplamak için, Hamilton yogunlugunun klasik faz uzayı degiskenleri xxx ve ppp yerinebirbirleriyle komute etmeyen kuantum mekaniksel faz uzayı operatörleri olan (PPP, RRR) yebaglı oldugunu düsünelim. Bu durumda Foldy-Wouthuysen dönüsümleri uniter olmaktançıkarlar ve uniterligi saglamak için birinci mertebe h içeren bir terim eklemek gerekir. Budurumu söyle ifade edebiliriz:

UFW →UFW +X UFW .

Buradaki X Berry ayar alanlarına baglıdır. Sonraki asamada ise yeni dönüsümkullanılarak birinci derece h mertebesinde olan tüm terimleri içeren yarı klasik blokkösegen Hamilton yogunlugu tam olarak hesaplanmıs olur. Fakat blok kösengenlestirmeislemi sırasında Hamilton yogunlugunun baglı oldugu faz uzayı degiskenlerinin cebrinon-komutatif olur. Hesaplanmıs olan Hamilton yogunlugu tezimin bir sonrakikısımlarındaki hareket denklemlerinin çözümünde ve spinin zaman içindeki degisimininhesabında kullanılacaktır.

Tezin ikinci kısmında, yarıklasik yöntemde tek parçacık durumunu ifade edebilmekiçin Dirac denkleminin pozitif enerji çözümlerinden olusan bir dalga paketi kuracagım.Kurulan dalga paketi ile yarıklasik birinci derece Lagrange yogunluguna karsılık gelen η

bir-form elde edilir.

Sonraki bölümde ise bir-form η aracılıgıyla simplektik iki-form ω olusturulur. Bu spinindisleri ile yazılan bir matristir. Berry ayar alanı içeren ω Hamilton formalizmini eldeetmek için kulanılır.

Simplektik iki-form üzerinde diferansiyel yöntem olan matris degerli vektör alanının iççarpım islemi yapılarak hareket denklemleri elde edilmis olur. Dirac parçacıklarınınhızları için hareket denklemi çözümlerine ise yine bir diferansiyel form yöntemi olanLie türevi islemiyle ulasılır. Bu bölümde, hareket denklemi çözümlerini iki ayrıislemin sonuçlarını karsılastırarak hesapladım. Bunların ilki, 3+ 1 uzayzaman boyutuiçin tanımlanan hacim formun Pfaffian matrise baglı olarak yazılmasıdır. Pfaffianmatris bir kare matris için determinantının karekökü olarak tanımlanır. Diger yol

xviii

ise, yine 3 + 1 uzayzaman boyutu için, hacim formu simpletik iki-forma baglı olaraktanımlamaktır. Hareket denklemi çözümlerine, iki farklı terimle belirlenmis hacimformun Lie türevlerinin hesaplanması ve elde edilen denklemlerin karsılastırılmasıylaulasılır.

Kütleli, spin-1/2 parçacıklar ile çalısmama ragmen kullandıgım yarıklasik yöntemdespine karsılık gelen klasik serbestlik derecesi yoktur. Sistemdeki hız ifadeleri matrisdegerli olduklarından buldugum parçacık akım yogunlugu da matris degerlidir. Buradakiönemli nokta, serbest Dirac parçacıgı için spin korunumlu bir büyüklük olmamasınaragmen helisite operatörü korunumlu bir büyüklüktür. Bu nedenle helisite operatörünükullanarak bir spin akısı türetilebilir. Dagılım fonksiyonu ve süreklilik denklemibaslıgındaki kısımda, parçacıkları sag elli ve sol elli olmak üzere iki kısma ayırarakdagılım fonksiyonunu elde etmek istiyorum. Bu nedenle dagılım fonksiyonunu kösegenolarak yazmak için sistemimdeki bazı degistirerek helisite bazına geçecegim.

Daha sonra ise helisitenin kösegen oldugu bazda kurdugum dagılım fonksiyonunu tersdönüsüm ile ilk bazda ifade edecegim. Kütleli fermiyonlar için sag elli ve sol elliparçacıklar dengede oldugundan dagılım fonksiyonunun ilk bazda da kösegen olarak ifadeedilebilecegini gösterecegim. Bu bölümde son olarak ise süreklilik denklemini türeterekkütleli fermiyonların süreklilik denklemini sagladıgını gösterecegim.

Önceki bölümlerde Dirac parçacıklarının hızları için hareket denklemlerinin çözümlerini,kütleli fermiyonların dagılım fonksiyonunu ve süreklilik denklemini elde ettim. Buislemlerin ardından ise Dirac parçacıklarının hızları için bulunan hareket denklem-lerindeki tüm ifadeleri helisite bazında yazarak kütlesiz limitini hesaplayacagım. Böylecekütlesiz fermiyonlar için hareket denklemlerinin çözümünü elde edecegim. Ayrıca,kütlesiz fermiyonların parçacık akısını ve süreklilik denklemlerini bulacagım. Diracparçacıklarının aksine, süreklilik denklemini saglamadıklarını ve anomaliye sahipolduklarını gösterecegim.

Dalga paketi yöntemiyle Dirac parçacıkları için elde edilen hareket denklemiçözümlerindeki hız ifadeleri Berry egriligi terimlerini içeren "anormal hız" terimlerinesahiptir. Oysa, Dirac parçacıklarının kovaryant formalizmi ile elde edilen hareketdenklemlerinde Thomas presesyonu nedeniyle anormal hız terimleri yoktur. Thomaspresesyonu spin matrisinin relativistik olmayan hareket denklemlerinde bir kinematikdüzeltme terimi olarak bulunmustur. Bununla birlikte Thomas presesyonu faz uzayıdegiskenlerinin hareket denklemlerine katkı saglamalıdır. Diferansiyel form ve dalgapaketi formalizmi ile kurdugum relativistik olmayan sistemin, Thomas dönmesi katkısınıiçermemesi beklenen bir durumdur. Bu durumu düzeltmek için yarıklasik formalizmineThomas dönmesi yerlestirilmelidir.

Thomas presesyonu bir Lorentz ötelemesinin ard arda uygulanan iki Lorentz ötelemesive dönme ifadesi cinsinden yazılmasından kaynaklanır. Buradaki dönme ifadesineaynı zamanda Thomas dönmesi de denilmektedir. Bu sayede Dirac denkleminikullanmadan elektronun spinin zamana göre degisiminin ifadesi dogru bir sekildehesaplanabilmektedir.

Kullandıgım yarı klasik yöntemin Thomas presesyonun katkısını hesaplamak için çokuygun oldugunu görecegim. Thomas dönmesini ve sistemime nasıl bir katkı verdiginibularak, momentumun yüksek mertebe katkısını ihmal ettigimde anormal hız terimlerininkayboldugunu gösterecegim. Bu sonuç ilk defa bu tezde bulunmustur.

Thomas presesyonu katkısının incelenmesinin ardından spin matrislerinin zaman içindekidegisimi hesaplanacaktır. Kullandıgım yöntemde spin matrisleri Pauli spin matrisleri

xix

ile ifade edilmektedirler ve faz uzayı yöntemi spinin hareketlerini belirlemez. Bunedenle spin hareket denklemleri baska yöntemlerle bulunmalıdır. Bunun içinblok kösegen Hamilton yogunlugunu bulmakta kullandıgım Gosselin-Berard-Mohrbachyöntemini kullanacagım. O formalizmde uniter dönüsüm sonrası faz uzayı islemcilerinon-komütatif olurlar. Dolayısıyla Pauli matrisleri ile de komüte etmezler. Bu katkılargöz önüne alındıgında Gosselin-Berard-Mohrbach yöntemiyle elde ettigim sonucun,elektromanyetik alanda hareket eden elektronun spininin zaman içindeki degisimini verenBargmann-Michel-Teledgi denklemiyle aynı sonucu verdigini gösterecegim.

Tezin son bölümünde ise elde ettigim sonuçlar ve bazı uygulamaları tartısılmıstır.

xx

1. Introduction

Dirac equation which describes massive spin-1/2 particles possesses either positive or

negative energy solutions described by spinors. However, to get a well defined one

particle interpretation a wave packet build of only positive energy plane wave solutions

should be preferred. Employing this wave packet one can obtain a non-relativistic

semiclassical dynamics which may be useful to have a better understanding of some

quantum mechanical phenomena. In the massless limit Dirac equation yields chiral

particles called Weyl particles. Recently it has been shown that chiral semiclassical kinetic

theory can be formulated embracing the anomalies due to the external electromagnetic

fields in 3 + 1 dimensions [1, 2]. This remarkable result was extended to any even

dimensional space-time by making use of differential forms in [3] by introducing some

classical variables corresponding to spin. Although in [3] non-Abelian anomalies have

been incorporated into the particle currents successfully, the solutions of phase space

velocities in terms of phase space variables were missing. In [4] a complete description

of the chiral semiclassical kinetic theory in the presence of the external electromagnetic

fields, in any even space-time dimension was established by introducing a matrix valued

symplectic two-form, without introducing any classical variable corresponding to spin

degrees of freedom. In this formalism, although the classical phase space variables are

the ordinary ones, the velocities arising from the equations of motion are matrix valued. It

has been shown in [5] that this matrix valued symplectic two form can be derived within

the semiclassical wave packet formalism [6, 7].

One of the main ingredients of these semiclassical formalisms of chiral kinetic theory is

the Berry gauge field whose field strength yields the quantum mechanical phase factor

known as the Berry phase summarized in Appendix A. To study dynamics starting with

a first order Lagrangian, the related Hamiltonian should be provided. In the development

of the chiral kinetic theory the Hamiltonian was taken as the positive relativistic energy

of the free Weyl Hamiltonian, i.e. the magnitude of the momentum vector. However,

later it was shown that the adequate Hamiltonian should contain all the first order terms

in Planck constant [8] which can be attained by employing the method introduced in

1

[9]. Independently, the same Hamiltonian was conjectured in [10] to restore the Lorentz

invariance of the semiclassical chiral theory. To obtain this semiclassical Hamiltonian

one first has to derive the Hamiltonian of the massive spin-1/2 fermion and then take the

massless limit.

Massive fermions also appear in condensed matter systems which were studied in terms

of wave packets in [11, 12]. The semiclassical kinetic theory of Dirac particles was also

discussed in [13], where the Berry gauge fields described in a different basis and some

classical degrees of freedom have been assigned to spin. In the thesis the formalism given

in [4] is applied to attain the semiclassical kinetic theory of the Dirac particles. There

are some advantages of employing this method. First of all because of not attributing

any classical variable to spin but considering matrix valued quantities in spin space the

calculations can be done explicitly. The differential forms method of [4] provides us

the solutions of the equations of motion for the phase space velocities in terms of the

phase space variables straightforwardly. Thus the particle currents can readily be derived

except the difficulty interpreting the matrix elements of the distribution function. It is

a matrix in "spin indices" but spin components are not conserved. In contrary to spin,

helicity operator is a conserved quantity for the free Dirac particle. To consider the

massless limit and chiral currents when there is an imbalance of chiral particles, one

has to split the particles as right-handed and left-handed. Therefore introducing a change

of basis to the helicity basis would be appropriate. Moreover, I will show that within this

formalism one can study the relativistic corrections known as Thomas precession [14] in

a comprehensible manner.

Thomas precession stems from the fact that a Lorentz boost can be written as two

successive Lorentz boosts accompanied by a rotation which is called Thomas rotation.

This purely kinematic phenomenon is essential to obtain time evolution of electron’s

spin correctly without referring to the Dirac equation. Thomas precession should also

contribute to the equations of motion of phase space variables. In fact, due to the

Thomas precession the covariant formalism of the Dirac particles yield equations of

motion where anomalous velocity terms do not emerge [15, 16]. However, as we will

see the equations of motion derived within the wave packet formalism possess anomalous

velocity terms proportional to the Berry curvature. This would have been expected

because, our non-relativistic formalism is not aware of the Thomas rotation. Correction

due to the Thomas rotation should be installed in the formalism. I will show that to

2

take this correction into account the adopted formalism is very suitable: It contributes to

one-form obtained by semiclassical wave packet on the same footing as the Berry gauge

field. It yields the cancellation of the anomalous velocity terms up to higher order terms

in momentum.

For completeness, the semiclassical formalism which is adopted in this thesis should

be supported by an equation governing time evolution of spin. This will be attained

employing Gosselin-Berard-Mohrbach (GBM) method [9] which is also needed to

derive the semiclassical Hamiltonian. Thus, I briefly review this method which is a

generalization of the Foldy-Wouthuysen transformation [17] in Section 2. In Section 3

the one-form corresponding to the first order Lagrangian is obtained by the wave packet

composed of the positive energy plane wave solutions of the Dirac equation. Then the

related symplectic form is constructed and solutions of the equations of motion for the

velocities are established in Section 4. I clarified in Section 5 how to introduce a change

of basis where the helicity operator is diagonal. Employing distribution function in the

adequate basis I then can write the particle number density and the related current by the

velocities obtained in terms of the phase space variables in Section 4. In Section 6 the

massless case is established by constructing the helicity eigenstates explicitly. In Section

7 first a brief review of the Thomas rotation is presented . Then, I show how it contributes

to the one-form built with the semiclassical wave packet. We will see that up to higher

order terms in momentum it contributes as the the Berry gauge field but with an opposite

sign. I will show that GBM method can be used to derive time evolution of spin matrices

which coincides with the Bargmann-Michel-Teledgi equation [18] in Section 8. In the

last section the results obtained and some possible applications are discussed.

3

4

2. Semiclassical Diagonalization of The Dirac Hamiltonian

In this section, I present the semiclassical block diagonal Hamiltonian for the massive

fermions in the electromagnetic field using Gosselin-Berard-Mohrbach method [9]. This

method consists of several steps. To begin with, since I would like to write semiclassical

block diagonal Hamiltonian, the diagonalization matrix must be established up to the first

order terms in the Planck constant h. I will see that the terms which are linear in h are

related to the Berry gauge fields. Then, I apply the block diagonalization process. After

this, canonical variables become non-commutative.

Let us start with the Hamiltonian given by

H = H0 + e A0(xxx). (2.1)

where H0 is

H0(ppp− e aaa(xxx)) = βm+ααα · (ppp− e aaa(xxx)). (2.2)

This Hamiltonian describes the Dirac particle interacting with the external electromag-

netic fields E , BBB whose vector and scalar potentials are aaa(xxx) and A0(xxx). H is classical

because I deal with xxx and ppp which are commuting phase space variables.

I choose the representation of the αi and β matrices as

αi =

(0 σiσi 0

), β =

(1 00 −1

),

where σi are the Pauli spin matrices. I set the speed of light c = 1 and charge e < 0 for

electron.

The Hamiltonian which has commuting space and momentum variables is diagonalized

by the unitary Foldy-Wouthuysen transformation as

U(ppp− eaaa) =βH0 +E√2E(E +m)

, (2.3)

where E =√(ppp− e aaa)2 +m2 is the relativistic energy.

I would like to obtain a block diagonal Hamiltonian considering the quantum mechanical

phase space operators (PPP, RRR), which satisfy the canonical commutation relations

[Pi,R j] = ihδi j.

5

When one maps ppp → PPP and xxx → RRR, the Foldy-Wouthuysen transformation matrix

(2.3) ceases to be unitary. To restore unitary at the first order in h, one replaces the

Foldy-Wouthuysen transformation adding a term:

U(ppp− eaaa)→U(PPP− eaaa(R))+X U(PPP− eaaa(R)).

Here, X is of order h and it is not uniquely defined. One can write it as

X =i

4h[APi,ARi],

where APi and ARi are defined by

ARi = ih U∇PiU†, (2.4)

APi = ih U∇RiU†. (2.5)

I would like to emphasize that RRR dependence of U(PPP − eaaa(R)) is due to the

electromagnetic vector potential aaa(RRR).

The semiclassical block diagonalization transformation

HD = (U(PPP− eaaa(R))+X U)H0(PPP− eaaa(RRR))(U†(PPP− eaaa(R))+U† X†).

is exact up to order h terms. I compute this transformation and obtain the block diagonal

Hamiltonian

HD(πππ, rrr) = β E(rrr, πππ)+i

2hP[β E(rrr, πππ),ARi] APi− [β E(rrr, πππ),APi] ARi, (2.6)

where P denotes the projection on to the block diagonal terms and πππ = ppp− eaaa(rrr). The

transformed phase space variables are

rrr = P[U(PPP,RRR)RRR U†(PPP,RRR)] = RRR+P(AR), (2.7)

ppp = P[U(PPP,RRR)PPP U(PPP,RRR)] = PPP+P(AP). (2.8)

As one can easily observe these dynamical variables are non-commutative.

Then I explicitly get

HD(πππ, rrr) = β E− he(

mΣΣΣ ·BBB2E2 +

(BBB · πππ)(ΣΣΣ · πππ)2E2(E +m)

), (2.9)

where ΣΣΣ matrices are given by

Σi =

(σi 00 σi

). (2.10)

It is the semiclassical block diagonal Hamiltonian for the massive fermions in the external

electromagnetic fields.

6

3. Wave Packet Formalism of the First Order Lagrangian

In this section I would like to construct the semiclassical first order Lagrangian in terms

of the positive energy plane wave solutions of the Dirac equation [5]. I would like to use

differential forms, so that I start with deriving the one form η which will be used to write

the symplectic two-form needed for obtaining Hamiltonian dynamics.

A semiclassical wave packet formulation may provide us a well defined one particle

interpretation. Therefore, I define the wave packet consisting of positive energy solutions

of the Dirac equations. The position of the wave packet center in the coordinate space is

xxxc and the corresponding momentum is pppc. The wave packet can be defined by using the

positive energy solutions, uα(xxx, ppp); α = 1,2, as

ψxxx(pppc,xxxc) = ∑α

ξα uα (pppc,xxxc)e−ipppc·xxx

h .

For simplicity we deal with the constant coefficients ξα .

As I would like to attain the one-form η , it is defined through dS as

dS≡∫[dx]δ (xxxc− xxx)Ψ†

xxx (−ihd−HDdt)Ψxxx = ∑αβ

ξ∗αη

αβξβ , (3.1)

where HD is the block diagonal Hamiltonian found in (2.9). One can show that

dψx =∂ψx

∂xxxc·dxxxc +

∂ψx

∂ pppc·d pppc

= [∑β

ξβ

∂uβ (pppc,xxxc)

∂xxxce−ipppc·xxx

h ] ·dxxxc +[∑β

ξβ (∂uβ (pppc,xxxc)

∂ pppc− ihxxxuβ )e

−ipppc·xxxh ] ·d pppc.

Plugging this into (3.1) I obtain

dS = −

(∑α,β

ξα∗ihuα † ∂uβ

∂xxxcξβ

)·dxxxc−

(∑α,β

ξα∗ihuα † ∂uβ

∂ pppcξβ

)·d pppc

− ∑α,β

ξα∗Eαδ

αβξβ dt− xxxc ·d pppc.

Let us introduce the matrix valued Berry gauge fields

AAAαβ = ihu†(α)(pppc,xxxc)∂

∂ pppcu(β )(pppc,xxxc).

aaaαβ = ihu†(α)(pppc,xxxc)∂

∂xxxcu(β )(pppc,xxxc), (3.2)

7

which can be used to write dS as

dS=∫[dx]Ψ†

xxx (−ihd−HDdt)Ψxxx =∑αβ

ξ∗α

(−xxxc ·d pppcδ

αβ−aaaαβ ·dxxxc−AAAαβ ·d pppc−HDαβ dt

)ξβ .

Hence I calculated (3.1) and attained the one-form η in the general form as

ηαβ =−δ

αβ xxxc ·d pppc−aaaαβ ·dxxxc−AAAαβ ·d pppc−HDαβ dt.

Although we deal with the (3+ 1) dimensional space time, the derivation of the η is

independent of dimension.

8

4. Semiclassical Dynamics of the Dirac Particles

In this section instead of solving the Dirac equation in the presence of the electromagnetic

gauge potential aaa(xxx), I would like to consider the free solutions whose Hamiltonian is

given by (2.2) with the replacement ppp→ ppp+ eaaa(xxx). Then the positive energy solutions

will not have xxx dependence.

Therefore by renaming (xxxc, pppc)→ (xxx, ppp) and setting aaaαβ = 0, I obtain the following one

form

η = pidxi + eaidxi−Aid pi−Hdt (4.1)

where the repeated indices i = 1,2,3, are summed over. Here

H = HD(ppp)+ eA0(xxx),

where HD(ppp) is found by using (2.9) which is became 2× 2 matrices by the projection

operator onto the positive energy subspace I+.

HD(ppp) = E− he(

mσσσ ·BBB2E2 +

(BBB · ppp)(σσσ · ppp)2E2(E +m)

). (4.2)

I would like to derive the symplectic two-form ω to obtain the Hamiltonian formalism. I

adopt the definition of the symplectic two-form ω to be

ω = dη− iη ∧η .

Employing the one-form given in (4.1) it yields

ω = d pi∧dxi +Eidxi∧dt + fid pi∧dt− 12

Gi jd pi∧d p j +e2

Fi jdxi∧dx j

= d pi∧dxi +Eidxi∧dt + fid pi∧dt−G+ eF,

where Ei and fi are given by

Ei =−(

e∂ai

∂ t+

∂H∂xi

),

fi =

(−∂H

∂ pi− i

h[Ai,H]

).

9

The two-forms F and G are written in terms of the Berry field strength

Gi j =

(∂A j

∂ pi− ∂Ai

∂ p j− i[Ai,A j]

),

and the electromagnetic field strength tensor

Fi j =

(∂a j

∂xi− ∂ai

∂x j

)as G = 1

2Gi jd pid p j and F = 12Fi jdxidx j.

The equations of motion can be found calculating the following:

ivω = 0. (4.3)

where iv is the interior product of the vector field

v =∂

∂ t+ ˙xi

∂

∂xi+ ˙pi

∂

∂ pi. (4.4)

( ˙xi, ˙pi) are the matrix-valued time evolutions of the phase space variables (xi, pi). The

equation of motion is given by making use of (4.3) and (4.4):

˙xi = fi + ˙pc Gci,

˙pi = Ei − e ˙xc Fci.

The solutions of the equation of motions for the velocities of the Dirac particle are attained

thanks to Lie derivative of the volume form as I will derive in the following. The volume

form for (3+1) spacetime dimensions is given by

Ω =13!

ω ∧ ω ∧ ω ∧dt.

It can be written in terms of the canonical volume element of the phase space,

dV = dxi∧dx j∧dxk∧d pi∧d p j∧d pk,

Ω = ω1/2 dV ∧dt

where ω1/2 is the Pfaffian of the following matrix,(Fi j −δi jδi j Gi j

)Obviously, ω1/2 is still a matrix in the (α,β ) indices. For an n× n matrix, the Pfaffian

matrix is described as the square root of its determinant. It is basically given by

10

completely antisymmetric tensor. However, I do not need its explicit definition as we

will see.

Time evolution of the volume form Ω can be found by calculating the Lie derivative

associated with v :

LvΩ = (ivd +div)ω1/2.

It can be calculated into the two different ways. Firstly, the Lie derivative of the volume

form can be written in terms of the Pfaffian,

LvΩ = (ivd +div)(ω1/2dV ∧dt)

=

(∂ω1/2

∂ t+

∂ (ω1/2 ˙xi)

∂xi+

∂ (ω1/2 ˙pi)

∂ pi

)dV ∧dt (4.5)

The second way is

LvΩ = (ivd +div)(13!

ω3∧dt)

=13!

dω3 (4.6)

To calculate it let me write explicitly

dω3 =

∂ω3

∂ tdt +

∂ω3

∂xidxi +

∂ω3

∂ pid pi.

where ω3 is,

ω3 = d pi∧dxi∧d p j∧dx j∧d pk∧dxk +3Eidxi∧d p j∧dx j∧d pk∧dxk∧dt

− 6F ∧Gd p j∧dx j +6 fiF ∧d pi∧dx j∧d p j∧dt−6EiG∧d p j∧dx j∧dxi∧dt

− 6EiF ∧G∧dxi∧dt−6 fiF ∧Gd pi∧dt.

dω3 can be written

dω3 = [

∂

∂ t(3!+

64

e(FmnGmn−FnmGmn))

+∂

∂xi(−3.2! fi +6EnGan−6e flFklGak−

62

e fiFklGkl)

+∂

∂ pi(2!3Ei +6e fnBan−6eElGklFak−

62

eEiGklFkl)]dV ∧dt, (4.7)

where Fi j is written by

Fi j = εi jkBk, (4.8)

and Gi j is given as

Gi j = εi jkGk. (4.9)

11

When (4.7) is substituted in (4.6) and compared with (4.5), we find the solutions of the

equations of motion in general form [4]:

ω1/2 = 1+ e BBB ·GGG (4.10)

ω1/2 · ˙xi =− fi +(E ×GGG)i− eBi( fff ·GGG) (4.11)

ω1/2 · ˙pi = Ei− ( fff × eBBB)i +Gi(E · eBBB) (4.12)

In order to write the solution of the equations of motion for Dirac particle explicitly, let

me first give the positive solutions for H0(ppp) = ααα · ppp+βm. They can be obtained as

Uα(ppp) =U(ppp)u0α .

where the rest frame solutions are

u01 =

1000

, u02 =

0100

.

U(ppp) is the Foldy-Wouthuysen transformation:

U(ppp) =βH0(ppp)+E√

2E(E +m)

=

m+E(√

2E(E+m)I σσσ ·ppp√

2E(E+m)

− σσσ ·ppp√2E(E+m)

m+E√2E(E+m)

I

. (4.13)

The Berry gauge field can be found by using Foldy-Wouthuysen transformation matrix.

Ai = ih I+ U(ppp)∂U†(ppp)

∂ piI+, (4.14)

Then I acquire,

AAA =−hσσσ × ppp

2E(E +m). (4.15)

Hence, by substituting (4.15) in the definition of the Berry curvature (4.9), I get

GGG =− h2E3 m

(σσσ +

ppp(σσσ · ppp)m(m+E)

). (4.16)

Observe that the covariant derivative of the Berry curvature vanishes,

DiGi =∂Gi

∂ pi− i

h[Ai,Gi] = 0. (4.17)

The first term can be calculated as

∂Gi

∂ pi=

hmE4(E +m)

σσσ · ppp.

12

The latter term is obtained by using (4.15) and (4.16):

ih[Ai,Gi] =

−im2E3 [Ai,σ j](δi j +

pi p j

m(m+E))

=hm

2E4(E +m)(σσσ · pppδi j−σi p j)(δi j +

pi p j

m(m+E))

=hm

E4(E +m)σσσ · ppp.

Hence I established the result reported in (4.17).

I explicitly obtain the velocities (4.11), (4.12), weighted by (4.10) as

ω1/2 = 1− eh

2E3 m(

σσσ ·BBB+ppp ·BBB(σσσ · ppp)m(m+E)

), (4.18)

ω1/2 ˙xi =− fi− εi jkh

2E3 m(

Eiσk +Ei pk(σσσ · ppp)m(m+E)

)− e Bi( fff ·GGG), (4.19)

ω1/2 ˙pi = Ei− e( fff ×BBB)i− eh

2E3 m(

σi +pi(σσσ · ppp)

m(m+E)

)(E ·BBB). (4.20)

where fi is written by

fi =−pi

E+ he

(− pi (σσσ ·BBB)

E4 +Bi(σσσ · ppp)(E−m)

2E3(E +m)+m σi

σσσ ·BBBE3(E +m)

− pi(σσσ · ppp)((BBB · ppp))

2E4(E +m)

).

13

14

5. Distribution Function and Continuity Equation

I study spin-12 massive particles where the velocities are matrices in “spin spaces”. So

that one should consider a matrix valued distribution function. For a Dirac particle spin

is given by the Σ matrices defined in (2.10). For the semiclassical wave packet composed

of the positive energy solutions it is projected on to

uα †ΣΣΣ uβ = σσσ

αβ .

They do not commute with the free Dirac Hamiltonian so that they are not conserved in

time. However the helicity operator

λαβ = uα † (

ΣΣΣ · pppp

) uβ =σσσ · ppp

p

commutes with the free Dirac Hamiltonian.

It is appropriate to split up the particles as right-handed and left-handed. This can be

performed in the basis where the helicity λ is diagonal. In order to establish the diagonal

basis, I use the spherical coordinates given by,

px = psinθ cosφ

py = psinθ sinφ

pz = pcosθ .

λ is diagonalized by the unitary matrix:

R =

(cos(θ

2 ) −sin(θ

2 )e−iϕ

sin(θ

2 )eiϕ cos(θ

2 )

).

Thus the helicity basis is defined by

φ = R u.

Now, the distribution function can be written in the diagonal basis as

fφ =

(fR 00 fL

).

15

The distribution function in the initial basis can be obtained by using transformation as

fu = R fφ R† =

(cos(θ

2 ) sin(θ

2 )e−iφ

−sin(θ

2 )eiφ cos(θ

2 )

)(fR 00 fL

)(cos(θ

2 ) −sin(θ

2 )e−iφ

sin(θ

2 )eiφ cos(θ

2 )

)=

(cos2(θ

2 ) fR + sin2(θ

2 ) fLsin(θ)

2 e−iφ ( fR− fL)sin(θ)

2 eiφ ( fR− fL) cos2(θ

2 ) fR + sin2(θ

2 ) fL

).

Since the number of the right-handed and left-handed particles are equal for massive

spin-12 particles , I set fR = fL = f . So that the distribution function becomes

fu =

(fR 00 fL

)= f I. (5.1)

Let the distribution function,f, satisfy the collisionless Boltzmann equation:

d fdt

=∂ f∂ t

+∂ f∂xi

˙xi +∂ f∂ pi

˙pi = 0. (5.2)

In order to write the continuity equation, one should identify the particle number density

n(x, p, t), and the particle current density j(x, p, t). As all of these equations which I found

are matrix valued, I have to introduce an appropriate definition of the classical limit. This

can be done by taking their trace and define the classical velocities as

√W ≡ Tr[ω1/2],

√Wxi ≡ Tr[ω1/2 ˙xi],

√W pi ≡ Tr[ω1/2 ˙pi].

I can write the probability density function as ρ(x, p, t) =√

W f . Hence, the particle

number density and the particle current density are given by:

n(x, t) =∫ d3 p

(2π)3 ρ(x, p, t) =∫ d3 p

(2π)3 Tr[ω1/2] f ,

jjj(x, t) =∫ d3 p

(2π)3 ρ(x, p, t)xxx =∫ d3 p

(2π)3 Tr[ω1/2 ˙xxx] f , (5.3)

In order to attain the continuity equation, we need the Liouville equation. Equation (4.6)

can be written as

LvΩ = (ivd +div)(13!

ω3∧dt)

=12

dω ∧ ω2 (5.4)

where dω and ω2 are given by,

dω = −12

∂Gi j

∂ pkd pk∧d pi∧d p j

= − h mσσσ · pppE4(E +m)

d3 p, (5.5)

16

ω2 = EiFjkdxi∧dxk∧dxk∧dt

= 2E ·BBB d3x∧dt (5.6)

Hence,by making use of (5.5) and (5.6), (5.4) is rewritten as

LνΩ =12

dω ∧ω2 =

(h

mσσσ · pppE4(E +m)

E ·BBB)

dV ∧dt. (5.7)

Comparing (4.5) with (5.7), I can easily write

∂ω1/2

∂ t+

∂ (ω1/2 ˜xi)

∂xi+

∂ (ω1/2 ˜pi)

∂ pi= h

mσσσ · pppE4(E +m)

E ·BBB (5.8)

Here∫ d3 p

(2π)3∂ (ω1/2 ˜pi)

∂ pi= 0 because I suppose that there is no contribution from the

boundary of the momentum space.

In conclusion I reach the continuity equation using (5.3) and (5.8):

∂n∂ t

+∇∇∇ · jjj =∫ d3 p

(2π)3 Tr[hmσσσ · ppp

E4(E +m)E ·BBB f ] = 0.

One can also obtain the particle current

ji =∫ d3 p

(2π)3 Tr[ω1/2 ˙xi f ] =∫ d3 p

(2π)3

(pi fE

).

The Dirac particles satisfy the continuity equation, which has no anomaly.

17

18

6. Massless Fermions

I found the solution of the equations of motion for the phase space velocities in terms of

phase space variables in general form in section 4. In this part, I would like to study the

massless case. The massless Dirac equation can be written as two Weyl equations for the

right-handed and left-handed fermions. Therefore, the helicity basis is suitable to discuss

the massless case.

First of all in the Berry curvature (4.16) yields

GGG =−h bbbσσσ · ppp

p.

Here bbb = ppp2 p3 is the monopole field situated at p = 0: ∇∇∇ ·bbb = 2πδ 3(p)

Now, the Berry curvature is singular and instead of (4.17) it satisfies

∂Gφ i

∂ pi=−2π hσzδ

3(p).

When I change the bases from u to φ , the Berry curvature becomes

GGGφ = R† GGG R =−hbbb R† σσσ · pppp

R =−hbbb(

1 00 −1

).

Obviously, one can deal with the right-handed and left-handed fermions independently.

They produce the similar results. Then, let us deal only with the projection onto the

right-handed massless fermions. The projection matrix is,

PR =

(1 00 0

).

PR ωPR yields the scalar value

√ω ≡ 1− e hbbb ·BBB. (6.1)

The other solutions (4.19) and (4.20) obtain the massless limit:

√ω xi =− fi

φ − h εi jk E j bk + e h Bi fff ·bbb, (6.2)

√ω pi = Ei− e εi jk f j

φ Bk + e h bi (E ·BBB). (6.3)

19

where fi denotes the massless limit of fi.

In order to obtain the particle current density, one defines the probability density function

ρ(x, p, t) as√

ω f . f is distribution function for the right-handed fermions and satisfies

the collisionless Boltzmann equation (5.2). Thus the particle current density jjj is

ji =∫ d3 p

(2π)3

√ω xi f =

∫ d3 p(2π)3 (− f φ

i − h εi jk E j bk + e h Bi fff ·bbb) f .

where the last term where the current is parallel to the magnetic field is the chiral magnetic

effect term.

The continuity equation for massless particle can be calculated by using the Liouville

equation and making use of equation (4.5) and equation (5.4), the Liouville equation

possesses anomalies:(∂

∂ t

√ω +

∂

∂xi(√

ω xi)+∂

∂ pi(√

ω pi)

)= 2πδ

3(p)E ·BBB.

Thus I can write by using the definition of the probability density

∂ρ

∂ t+

∂ (ρ xi)

∂xi+

∂ (ρ pi)

∂ pi= 2πe f δ

3(p)E ·BBB.

The chiral anomaly is explained as non-conservation of the classically conserved chiral

current at the quantum level in quantum field theory. I show this phenomena by using the

probability function f , which satisfies the collisionless Boltzmann equation.

In conclusion I derive the anomalous continuity equation for massless fermions by using

the Boltzmann equation and (5.3) as

∂n(x, t)∂ t

+∇∇∇ · jjj =e

4π2 f (x, p = 0, t)E ·BBB.

The Berry monopole situated on the boundary |ppp| = 0 is responsible for the

non-conservation of the chiral particle current.

20

7. Thomas Precession

I would like to make clear how to take into account the Thomas precession [14] within the

wave packet formalism. To this end let us first briefly recall Thomas precession following

[19].

The source of this phenomenon lies in the fact that if one would like to write a Lorentz

boost as two successive Lorentz boosts, one also should rotate the coordinates with an

angle depending on the velocities. This rotation yields an angular velocity known as the

Thomas precession.

Suppose that the Dirac particle is moving with the velocity vvv with respect to laboratory

frame at time t. Hence the particle’s co-moving frame denoted by the inertial spacetime

coordinates x′, is connected to the spacetime coordinates of the laboratory frame by the

Lorentz boost λboost(vvv) at time t:

x′ = λboost(vvv) x. (7.1)

Let the particle accelerate, so that it moves with the velocity vvv+ dvvv with respect to the

laboratory frame at time t + dt. Then at time t + dt the co-moving coordinate frame

coordinates x′′ will be connected to the laboratory frame by the Lorentz transformation

x′′ = λboost(vvv+dvvv) x. (7.2)

Let me write the connection between the two co-moving frame coordinates x′ and x′′ as

x′′ = λT x′.

Inspecting (7.1) and (7.2) the transformation λT can be written as

λT = λboost(vvv+dvvv) λboost(− vvv). (7.3)

The Lorentz boost λboost(vvv+ dvvv) can be separated into two successive Lorentz boosts

accompanied by the rotation R(dθθθ)

λboost(vvv+dvvv) = R(dθθθ) λboost(dvvv) λboost(vvv), (7.4)

21

where the angle of the rotation is

dθθθ =γ2

γ +1vvv×dvvv. (7.5)

Here γ = 1√1−v2 is the relativistic factor

Therefore by plugging (7.4) into (7.3)

λT = R(dθθθ) λboost(dvvv). (7.6)

In the nonrelativistic systems successive co-moving frames of the Dirac particles are

connected by only boosts without any rotations. Therefore, x′′′ coordinates of the frame

moving with the velocity vvv+ dvvv at time t + dt will be obtained from the system moving

with the velocity vvv at time t only with the boost λboost(dvvv) without any rotation:

x′′′ = λboost(dvvv) x′.

Then by making use of (7.1), (7.2) and (7.3) the coordinates of the co-moving frame, x′′′,

are written in terms of the laboratory frame coordinate as

x′′′ = R(−dθθθ) λboost(vvv+dvvv) x. (7.7)

In the wave packet formalism one deals with the group velocity

vvv≡ ∂E∂ ppp

=pppE. (7.8)

Since γ = Em , by plugging (7.8) into (7.5) one gets

dθθθ =ppp×d ppp

m(E +m).

Let the laboratory frame and co-moving reference frames coincide at the time t = 0 when

the particle is at rest. Hence the solution of the Dirac equation in laboratory frame is u(0).

Now in the nonrelativistic system, one deals with,

du(ppp)NR = u′′′(ppp+d ppp)−u′(ppp) = R(−dθθθ) λboost(vvv+dvvv)u(0)−λboost(vvv)u(0)

= R(−dθθθ)u(ppp+d ppp)−u(p)

Hence when one considers the Thomas precession, the one-form η of the nonrelativistic

wave packet formalism will lead to

ihu†(ppp)du(ppp)NR = ihu†(ppp) [R(dθθθ)u(ppp+d ppp)−u(ppp)] (7.9)

22

Rotation of the spinors is given as

R(dθθθ) = 1− i4

σi j dωi j

Here σi j and dωi j are written as

σi j =i2[γi , γ j] = εi jk

(σk 00 σk

),

dωi j = ε

i jm dθ m =−εi jm dθ

m.

I can write the rotation as

R(dθθθ) = 1+D(dθθθ).

where D(dθθθ) is

D(dθθθ) =

(σσσ ·dθθθ 0

0 σσσ ·dθθθ

).

(7.9) yields

ihu†(ppp)du(ppp)NR = ihu†(ppp)∂u(ppp)

∂ ppp·d ppp+ ihu†(ppp)D(−dθθθ)u(ppp),

keeping only the first order terms in d ppp and dvvv = d pppm The first term is the Berry gauge

field calculated in (4.15). The Thomas precession term can be shown to be

ihu†(ppp)D(−dθθθ)u(ppp) =(hσσσ × ppp)

4m2 ·d ppp. (7.10)

Hence when the Thomas correction is considered the velocities can be read from (4.18),

(4.19) and (4.20) by the replacement of the Berry gauge fields (4.15) as

AAAT =−(hσσσ × ppp)4m2 ·d ppp +

(hσσσ × ppp)4m2 ·d ppp = 0

up to p2 terms, which were ignored in the calculation of the Thomas corrections.

Therefore, I conclude that the anomalous velocity terms disappear when the Thomas

rotation is considered.

23

24

8. Time Evolution of Spin

I would like to write the semiclassical equation governing time evolution of the spin for

the semiclassical Dirac particle in the electromagnetic field. However, in my formalism,

there is no classical degrees of freedom corresponding to the spin of the particle. The spin

for the wave packet composed of the positive energy solutions is σσσ . Therefore, I use the

σσσ matrices which corresponds to spin and use the semiclassical Hamiltonian (4.2).

However as it has been discussed in sec. 2, the new dynamical variables rrr and ppp are

non-commutative and depend on spin matrices. Thus I get

dσi

dt=

1i h

[σi,H(rrr+)] =1

i h

([σi,H]+ [σi, rrr+]

∂H∂ rrr+

), (8.1)

where ∂H∂ rrr+

=−eEEE and rrr+ is defined by (2.7) projecting on positive energy solutions:

rrr+ = RRR+(I+ AAA I+) = RRR− hσσσ × ppp

2E(E +m).

Thus the last term in (8.1) can be written as

dσi

dt=

1i h

[σi,H]−e E j

i hΘ

σ ri j . (8.2)

where Θσ r+i j is

Θσ ri j = [σi, r+ j] = [σi,A j] =

−ihE(E +m)

[(σσσ · ppp) δi j−σ j pi].

In the Hamiltonian (4.2) I ignore higher order terms in p2:

H = E− e h σσσ ·BBB2E

. (8.3)

By plugging (8.3) into 8.2 I get

dσσσ

dt=

eE

σσσ ×BBB+e

E(E +m)[Ei (σσσ · ppp) − pi (σσσ ·EEE)],

where the last term can be written as

Ei (σσσ · ppp) − pi (σσσ ·EEE) = εi jm εmkl σ j Ek pl. (8.4)

Therefore by using (8.4) and setting γ = Em , one reaches the time evolution of the spin

dσσσ

dt=

em

σσσ ×[

1γ

BBB+1

γ +1EEE× vvv

]. (8.5)

Observe that (8.5) is the BMT equation as composed in Jackson . [18, 19]

25

26

9. Results and Discussion

Semiclassical kinetic theory of massive spin-1/2 particles are studied within the method

introduced in [4]. The main ingredients are the symplectic-two form which is a matrix in

the spin indices related to the positive energy solutions of the Dirac equation. These

solutions constitute the wave packet which leads to the semiclassical approximation.

The block diagonal Hamiltonian including all terms which are at the first order in

Planck constant obtained from the Dirac Hamiltonian in the presence of the external

electromagnetic fields is presented. By projecting it on the positive energy subspace I

obtain the Hamiltonian which is used to define the semiclassical symplectic two-form

which is the starting point of the formulation. Differential forms are used to define the

semiclassical Hamiltonian dynamics of Dirac particles. The solutions of the equations of

motion for the phase space velocities in terms of the phase space variables are derived.

Then I used them to define the continuity equation of the particle number density and

particle number current. To achieve it one has to define distribution function adequately.

This is possible in the basis where the helicity operator is diagonal. Therefore, I performed

a change of basis such that the helicity operator becomes diagonal. This is also needed to

obtain the massless limit. I showed that the massless limit yields the continuity equation

with an anomaly term and also to the particle current yielding the chiral magnetic effect

as expected.

Thomas precession correction needed in the non-relativistic formulation is studied within

the wave packet formalism. I showed that up to higher order terms in momentum it sweeps

out the contribution coming from the Berry gauge field. This coincides with the results

obtained in relativistic formulation of Dirac particles. This is a result which is reported

for the first time in physics literature. The method of introducing the Thomas precession

correction which I presented is valid in general. It can be applied to other semiclassical

approaches of Dirac like systems where the underlying Hamiltonian of the theory is given

by Dirac like Hamiltonian as in some condensed matter systems.

The semiclassical kinetic theory formulation of Dirac particles and obtaining the massless

limit by explicitly constructing the suitable basis which are developed in this thesis can

27

be generalized to systems which have some other interaction terms in the Hamiltonian.

Obviously in the development of the semiclassical kinetic theory the next important step

is to switch-on interactions in the Boltzmann equation which are ignored in my study.

28

REFERENCES

[1] Son, D.T. and Yamamoto, N. (2012). Berry Curvature, Triangle Anomalies, and theChiral Magnetic Effect in Fermi Liquids, Physical Review Letters, 109(18),181602, 1203.2697.

[2] Stephanov, M.A. and Yin, Y. (2012). Chiral Kinetic Theory, Physical Review Letters,109(16), 162001, 1207.0747.

[3] Dwivedi, V. and Stone, M. (2014). Classical chiral kinetic theory and anomalies ineven space-time dimensions, Journal of Physics A Mathematical General,47(2), 025401, 1308.4576.

[4] Dayi, O.F. and Elbistan, M. (2014). A Semiclassical Formulation of the ChiralMagnetic Effect and Chiral Anomaly in Even d+1 Dimensions, ArXive-prints, 1402.4727.

[5] Dayi, Ö.F. and Yunt, E. (2014). Relation Between the Spin Hall Conductivity and theSpin Chern Number for Dirac-like Systems, ArXiv e-prints, 1408.1596.

[6] Sundaram, G. and Niu, Q. (1999). Wave-packet dynamics in slowly perturbedcrystals: Gradient corrections and Berry-phase effects, Phys. Rev. B, 59,14915–14925, cond-mat/9908003.

[7] Culcer, D., Yao, Y. and Niu, Q. (2005). Coherent wave-packet evolution in coupledbands, Phys. Rev. B, 72(8), 085110, cond-mat/0411285.

[8] Manuel, C. and Torres-Rincon, J.M. (2014). Chiral transport equation from thequantum Dirac Hamiltonian and the on-shell effective field theory, PhysicalReview D, 90(7), 076007, 1404.6409.

[9] Gosselin, P., Bérard, A. and Mohrbach, H. (2007). Semiclassical diagonalization ofquantum Hamiltonian and equations of motion with Berry phase corrections,European Physical Journal B, 58, 137–148, hep-th/0603192.

[10] Chen, J.Y., Son, D.T., Stephanov, M.A., Yee, H.U. and Yin, Y. (2014). LorentzInvariance in Chiral Kinetic Theory, Physical Review Letters, 113(18),182302, 1404.5963.

[11] Chang, M.C. and Niu, Q. (2008). Berry curvature, orbital moment, and effectivequantum theory of electrons in electromagnetic fields, Journal of physics.Condensed matter, 20, 193202, 0953-8984.

[12] Chuu, C.P., Chang, M.C. and Niu, Q. (2010). Semiclassical dynamics and transportof the Dirac spin, Solid State Communications, 150, 533–537, 0911.4760.

[13] Chen, J.W., Pang, J.y., Pu, S. and Wang, Q. (2014). Kinetic equations formassive Dirac fermions in electromagnetic field with non-Abelian Berryphase, Physical Review D, 89(9), 094003, 1312.2032.

29

[14] Thomas, L.H. (1926). The Motion of the Spining Electron, Nature, 117, 514.

[15] Stone, M., Dwivedi, V. and Zhou, T. (2015). Berry Phase, Lorentz Covariance, andAnomalous Velocity for Dirac and Weyl Particles, Physical Review D, 91(2),025004, 1406.0354.

[16] Duval, C. and Horváthy, P.A. (2004). Anyons with anomalous gyromagnetic ratioand the Hall effect, Physics Letters B, 594, 402–409, hep-th/0402191.

[17] Foldy, L.L. and Wouthuysen, S.A. (1950). On the Dirac Theory of Spin 1/2 Particlesand Its Non-Relativistic Limit, Phys. Rev., 78, 29.

[18] Bargmann, V., Michel, L. and Telegdi, V.L. (1954). Precession of the Polarization ofParticles Moving in a Homogeneous Electromagnetic Field, Physical ReviewLetters, 2, 435.

[19] Jackson, J.D. (1975). Classical Electrodynamics, John Wiley, second Edition.

[20] Sakurai, J.J. (1967). Advanced Quantum Mechanics, Addison-Wesley.

[21] Bohm, A., Mostafazadeh, A., Koizumi, H., Nui, Q. and Zwanziger, J. (2003). TheGeometric Phase in Quantum System, Springer-Verlag.

30

APPENDICES

APPENDIX A :Berry Gauge Fields

31

32

Berry Gauge Fields

Consider the Hamiltonian depending on the external variables such as RRR(R1,R2, ..., t)which rely on time dependence. The physical meaning can be thought as electricfield, magnetic field etc. When the parameters change slowly in a quantum mechanicalsystem, the quantum mechanical state returns to the initial condition only up to phasefactor after the cyclic evolution completed. The phase factor consist of the dynamicalphase in addition to a purely geometrical part. The geometrical phase term is calledthe Berry phase. Derivation of the Berry phase can obtained by the quantum adiabaticapproximation which is only related to slowly altering Hamiltonians. The Berry gaugefields are defined by using the Berry phase factor [20, 21]. To begin with, the eigenvalueequation is given by

H(R(t))|n(RRR(t))>= En(RRR(t))|n(RRR(t))> .

where R(T ) = R(0). This means that the cyclic evolution is closed.

In the adiabatic approximation the evolving state vector ψ(t) can be written in the basis|n(RRR(t))> as

ψ(t) = cn(t)|n(RRR(t))> .

The Schrodinger equation,

H(R(t))cn(t)|n(RRR(t))> = En(t)|n(RRR(t))>

= ih∂

∂ t[cn(t)|n(RRR(t))>] (A.1)

One can obtain

ih∂cn(t)

∂ t= cn(t)

[En(t)− ih < n(RRR)|∂ |n(R

RR)>∂ t

].

Then it becomes∫ cn(T )

cn(0)

dcn(t)cn

=∫ T

0

−ih

En(t ′)dt ′−∫ T

0< n(RRR)|∂ |n(R

RR)>∂ t ′

dt ′. (A.2)

where cn(t) is the phase factor including the dynamical and geometrical phase terms. Itcan be written as

cn(t) = e−ih∫ T

0 En(t ′)eih γn(t)

γn(t) is the Berry phase which is real valued

γn(t) =∫ T

0ih < n(RRR)|∂ |n(R

RR)>∂ t ′

dt ′ =∫ T

0ih < n(RRR)|[∇∇∇RRR|n(RRR)>] ·dRRR

As R(T ) = R(0) in the adiabatic approximation,

γn(t) =∮

ih < n(RRR)|[∇∇∇RRR|n(RRR)>] ·dRRR.

In the conclusion one reach the Berry gauge fields,

AAAn(RRR)≡ ih < n(RRR)|[∇∇∇RRR|n(RRR)>].

33

34

CURRICULUM VITAE

Name Surname: Eda Kılınçarslan

Place and Date of Birth: Izmir,Turkey, 26.01.1989

Adress:Physics Engineering Department, Faculty of Science and Letters, IstanbulTechnical University, TR-34469, Maslak–Istanbul, Turkey

E-Mail:kilincarslan@itu.edu.tr

B.Sc.:Istanbul Technical University, Physics Engineering Department

M.Sc.:Istanbul Technical University, Physics Engineering Department

35